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          群论基础及Pólya定理学习笔记
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            <div class="post-description">群论基础+波利亚定理，超长寒假的枯燥生活</div>

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        <h3 id="群"><a href="#群" class="headerlink" title="群"></a>群</h3><h4 id="定义"><a href="#定义" class="headerlink" title="定义"></a>定义</h4><p>&emsp;&emsp;一个非空集合 $G$ ，在它上面定义一个二元运算，对 $G$ 中的任意两元 $a, b$ ，记 $ab$ 为这个运算的结果（$ab$ 一定也在 $G$ 中）。这个运算满足下面三个性质：</p>
<p>  &emsp;&emsp;（1）<strong>结合律</strong>：对任何 $a,b,c$ ，有 $(ab)c=a(bc)$ ；</p>
<p>  &emsp;&emsp;（2）存在<strong>单位元</strong>：$G$ 中存在元  $e$ ，使得对任意 $G$ 中元 $a$ ，有 $ae = ea = a$ ，$e$ 叫做 $G$ 的<strong>单位元</strong>。</p>
<p> &emsp;&emsp; （3）存在<strong>逆元</strong>：对任何 $G$ 中元素 $a$ ，存在 $G$ 中元素 $a’$ 使得 $aa’ = a’a = e$ ，$a’$ 叫做 $a$ 的<strong>逆元</strong> 。</p>
<p>  &emsp;&emsp;这样的集合 $G$ 叫做一个群。</p>
<h4 id="一些性质及说明"><a href="#一些性质及说明" class="headerlink" title="一些性质及说明"></a>一些性质及说明</h4><ol>
<li>单位元唯一</li>
<li>$a$ 的逆元唯一</li>
<li>群 $G$ 满足<strong>消去律</strong>：如果 $ab = ac$ ，则 $b = c$ ；如果 $ba = ca$ ，则 $b = c$ 。</li>
<li>$|G|$ 叫做群 $G$ 的<strong>阶</strong>，即其所含元的个数。</li>
<li>$G$ 不一定满足<strong>交换律</strong>，若满足则这样的 $G$ 称作<strong>可换群</strong>。</li>
</ol>
<h4 id="拉格朗日定理"><a href="#拉格朗日定理" class="headerlink" title="拉格朗日定理"></a>拉格朗日定理</h4><h5 id="内容"><a href="#内容" class="headerlink" title="内容"></a>内容</h5><p>  &emsp;&emsp;任何有限群 $G$ 的子群 $H$ 的阶必是 $G$ 的阶的因子。</p>
<h5 id="证明"><a href="#证明" class="headerlink" title="证明"></a>证明</h5><p>  &emsp;&emsp;设 $H$ 是 $n$ 阶群 $G$ 的一个 $m$ 阶子群，$H$ 的元为 $b_1,b_2,…,b_m$ ，若 $m = n$ 则结论成立；</p>
<p>  &emsp;&emsp;若 $m \not= n$ ，则 $G$ 中必定存在 $a$ 不存在于 $H$ 中，于是可以看出 $ab_1,ab_2,..,ab_m$ 都不是 $H$ 中的元，否则运算可得 $a$ 在 $H$ 中，矛盾。我们把 ${ab_1,ab_2,…,ab_m }$ 这个集合称为 $H$ 的一个<strong>（左）陪集</strong>（注意，它不是群），记做 $aH$。此时若 $2m = n$ ，则 $G$ 为 $H$ 和 $aH$ 的并，结论成立。</p>
<p>  &emsp;&emsp;若 $2m \not= n$ ，则 $G$ 中一定存在 $a’$ 不存在于 $H$ 和 $aH$ 的并集中，此时相似地可以看出存在 $H$ 的陪集 $a’H$ ，且 $a’H$ 里的元不存在于 $H$ 和 $aH$ 的并集中，于是此时若 $3m = n$， 则 $G$ 为 $H \cup aH \cup a’H $ ，结论成立。</p>
<p>  &emsp;&emsp;于是，归纳可得结论成立，且 $G$ 一定是由 $H$ 和它的不同陪集合并而成。若 $n = km$ ，则 $k$ 称为子群 $H$ 的指数。</p>
<p><img src="https://i.loli.net/2020/03/12/wJgWq2zNeLAipRT.png" alt="图片.png" style="zoom:50%;" /></p>
<h4 id="两种常见的群"><a href="#两种常见的群" class="headerlink" title="两种常见的群"></a>两种常见的群</h4><h5 id="循环群"><a href="#循环群" class="headerlink" title="循环群"></a>循环群</h5><p>  &emsp;&emsp;PS：先定义一下<strong>乘方运算</strong>：$a^n$ 表示 $a$ 自乘 $n$ 次。另外，$a$ 的逆元也可记作 $a^{-1}$。</p>
<p>  给定 $G$ 中元 $a$ ，考虑由 $a$ 生成的全部子群，它由全部 $a^m$ 组成（$m$ 是任意整数），可记作</p>
<script type="math/tex; mode=display">
\langle a\rangle = \{\cdots,a^{-3},a^{-2},a^{-1},e,a,a^2,a^3,\cdots\}</script><p>若 $G = \langle a\rangle$ , $a$ 便是 $G$ 的单一个<strong>生成元</strong>， $G$ 便是个<strong>循环群</strong>。</p>
<p>  &emsp;&emsp;循环群由分为<strong>无限循环群</strong>和<strong>有限循环群</strong>，其中有限循环群即 $a^s = e$ 。</p>
<p>  &emsp;&emsp;令人兴奋的是，我们发现整数加群 $Z$ 就是一个无限循环群，而如果给它加上模数，则是有限循环群。</p>
<h5 id="二面体群"><a href="#二面体群" class="headerlink" title="二面体群"></a>二面体群</h5><p>  &emsp;&emsp;简单地说，二面体群由有限循环群和它的“镜像”组成。这类群可以用两个生成元刻画，一个叫 $a$ ，一个叫 $b$ ，它们都不是单位元 $e$ 。其中存在最小的正整数 $n$ 使得 $a^n = e$ ，$b$ 满足 $b^2 = e$ ，且 $ab = ba^{n-1}$ 。</p>
<p>  &emsp;&emsp;这个东西特别惊艳，可以从几何角度去理解！—— $a$ 就是绕中心转一个角度， $b$ 就是沿对称轴折叠！</p>
<p>  &emsp;&emsp;这样的群叫做 <strong>$2n$ 阶二面体群</strong>，通常记作 $D_n$ 。一个正 $n$ 边形的对称群就是个这样的群。我们通常从 $n = 3$ 开始定义 $D_n$ ，这时 $D_n$ 一定不是可换群。硬要说的话， $D_2$ 是可换群。</p>
<h4 id="置换群"><a href="#置换群" class="headerlink" title="置换群"></a>置换群</h4><h5 id="置换"><a href="#置换" class="headerlink" title="置换"></a>置换</h5><p>  &emsp;&emsp;就是指把 $N$ 个东西的排列次序<strong>调换</strong>，或者说，就是一个由这 $N$ 个东西组成的集自身之间的一一对应。我们可以把这 $N$ 个东西记作 $1、2、\cdots、N$ （<strong>不仅仅可以表示数字</strong>）。一个置换 $\sigma$ 可以表为</p>
<script type="math/tex; mode=display">
\dbinom{1\quad\quad 2 \quad\ \ \cdots \quad N }{\sigma(1)\quad \sigma(2) \quad \cdots \quad \sigma(N)}</script><p>&emsp;&emsp;$N$ 个不同东西的全部置换组成一个群，叫做 $N$ <strong>次对称群</strong>，记作 $S_N$ 。它的阶是 $N!$ 。单位元是 $\dbinom{1\quad 2 \quad \cdots \quad N }{1\quad 2 \quad \cdots \quad N}$ ，$\dbinom{1\quad\quad 2 \quad\ \ \cdots \quad N }{\sigma(1)\quad \sigma(2) \quad \cdots \quad \sigma(N)}$ 的逆元是 $\dbinom{\sigma(1)\quad \sigma(2) \quad \cdots \quad \sigma(N)}{1\quad\quad 2 \quad\ \ \cdots \quad N }$ 。</p>
<p>&emsp;&emsp;几何形体的对称群，通常可以表示为一个 $N$ 次对称群里面的某个子群。</p>
<h5 id="置换的圈分解"><a href="#置换的圈分解" class="headerlink" title="置换的圈分解"></a>置换的圈分解</h5><p>&emsp;&emsp;我们可以把一个置换 $\sigma$ 用一个图去表示，对每一个 $i$ ，从 $i$ 向 $\sigma(i)$ 画一条有向边，很容易发现这样的图一定是由若干个圈组成的。例如我们举一个例子：</p>
<p><img src="https://i.loli.net/2020/03/18/eGyHNIcCU4YBJj5.png" style="zoom:80%;"></p>
<p>&emsp;&emsp;它可以写成</p>
<script type="math/tex; mode=display">
\dbinom{1\ 2\ 3\ 4\ 5\ 6\ 7\ 8\ 9\ 10\ 11\ 12}{4\ 6\ 10\ 5\ 1\ 8\ 7\ 12\ 11\ 3\ 9\ 2} = (1, 4, 5)(2, 6, 8 ,12)(3, 10)(7)(9, 11)</script><p>&emsp;&emsp;一般而言，任何置换 $\sigma$ 都能唯一（不计次序）表成若干个圈的乘积，叫做 $\sigma$ 的圈分解。每个圈上的点数叫做他的圈长，圈长为 $1$ 的圈即是单位元置换，圈长为 $2$ 的圈叫做对换。</p>

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